Geodesic
Superdiffusions and positive solutions of non-linear partial differential equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 147-157 Cet article a éte moissonné depuis la source Math-Net.Ru

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By using super-Brownian motion, all positive solutions of the non-linear differential equation $\Delta u=u^\alpha$ with $1\alpha\leqslant 2$ in a bounded smooth domain $E$ are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case $\alpha=2$ was treated by B. Mselati in 2002. 
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E. B. Dynkin. Superdiffusions and positive solutions of non-linear partial differential equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 147-157. http://geodesic.mathdoc.fr/item/RM_2004_59_1_a9/

[1] E. B. Dynkin, “A probabilistic approach to one class of nonlinear differential equations”, Probab. Theory Related Fields, 89:1 (1991), 89–115 | DOI | MR | Zbl

[2] E. B. Dynkin, “A new relation between diffusions and superdiffusions with applications to the equation $Lu=u^\alpha$”, C. R. Acad. Sci. Paris Sér. I Math., 325:4 (1997), 439–444 | MR | Zbl

[3] E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[4] E. B. Dynkin, “On upper bounds for positive solutions of semilinear equations”, J. Funct. Anal., 210:1 (2004), 73–100 | DOI | MR | Zbl

[5] E. B. Dynkin, “New relations between diffusions and superdiffusions and their applications to differential equations”, Math. Res. Lett. (to appear)

[6] E. B. Dynkin, S. E. Kuznetsov, “Trace on the boundary for solutions of nonlinear differential equations”, Trans. Amer. Math. Soc., 350:11 (1998), 4499–4519 | DOI | MR | Zbl

[7] E. B. Dynkin, S. E. Kuznetsov, “Fine topology and fine trace on the boundary associated with a class of semilinear differential equations”, Comm. Pure Appl. Math., 51:8 (1998), 897–936 | 3.0.CO;2-0 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[8] E. B. Dynkin, S. E. Kuznetsov, “Poisson capacities”, Math. Res. Lett., 10:1 (2003), 85–95 | MR | Zbl

[9] E. B. Dynkin, S. E. Kuznetsov, “$\mathbb N$-measures for branching exit Markov systems and their applications to differential equations”, Probab. Theory Related Fields, 130:1 (2004), 135–150 | DOI | MR | Zbl

[10] A. Gmira, L. Véron, “Boundary singularities of solutions of some nonlinear elliptic equations”, Duke Math. J., 64:2 (1991), 271–324 | DOI | MR | Zbl

[11] S. E. Kuznetsov, “$\sigma$-moderate solutions of $Lu=u^\alpha$ and fine trace on the boundary”, C. R. Acad. Sci. Paris Sér. I Math., 326:10 (1998), 1189–1194 | MR | Zbl

[12] S. E. Kuznetsov, “An upper bound for positive solutions of the equation $\Delta u=u^\alpha$”, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 103–112 | DOI | MR | Zbl

[13] J.-F. Le Gall, “Les solutions positives de $\Delta u=u^2$ dans le disque unité”, C. R. Acad. Sci. Paris Sér. I Math., 317:9 (1993), 873–878 | MR | Zbl

[14] J.-F. Le Gall, “A probabilistic Poisson representation for positive solutions of $\Delta u=u^2$ in a planar domain”, Comm. Pure Appl. Math., 50:1 (1997), 69–103 | 3.0.CO;2-E class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[15] M. Marcus, L. Véron, “The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case”, Arch. Rational Mech. Anal., 144:3 (1998), 201–231 | DOI | MR | Zbl

[16] M. Marcus, L. Véron, “The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case”, J. Math. Pures Appl. (9), 77:5 (1998), 481–524 | MR | Zbl

[17] M. Marcus, L. Véron, “Capacitary estimates of solutions of a class of nonlinear elliptic equations”, C. R. Math. Acad. Sci. Paris, 336:11 (2003), 913–918 | MR | Zbl

[18] B. Mselati, Classification et représentation probabiliste des solutions positives de $\Delta u=u^2$ dans un domaine, Thése de Doctorat de l'Université Paris VI, 2002